# coarsening coefficient matrixes (A2h, A4h...) for geometric multigrid method in 2-D/3-D

I am learning about multigrid methods from the textbook section 6.3 Multigrid Methods, which shows a geometric multigrid algorithm for 1-D examples in detail, including how to build restriction/interpolation matrices (i.e., eqs.1, 3) and how to get a coarse coefficient matrix (eq. 6).

I realize that these matrices and way of coarsening the coefficient matrix are important for multigrid methods.

But for the 2-D case, only the ways of getting restriction and interpolation matrices are shown in the book, not the coefficient matrix for coarse grid (i.e., $$A_{2h}$$, $$A_{4h}$$, etc.).

This missing link is barrier for me to go further to learn multigird methods.

Can anybody help me?

• Gilbert Strang is a known name in many fields and to me, this document is intended to be just a starter document - more like a lecture note. Volker John also has a good set of lecture notes that you might be interested in: wias-berlin.de/people/john/LEHRE/MULTIGRID/multigrid.pdf . However, in my opinion, you should ask for references to learn about multigrid methods. I am pretty sure there are many great books out there about it. Jul 19, 2020 at 16:55

For geometric multigrid, the $$\mathbf A_{2h}$$ and $$\mathbf A_{4h}$$ (etc) matrices are just discretizations of the same PDE on coarser grids.
For instance, if your original $$\mathbf A_{h}$$ was a finite difference approximation of the laplacian on a 64 point grid, then $$\mathbf A_{2h}$$ would be a finite difference approximation of the laplacian on a 32 point grid, etc. You should also see some sort of relationship where $$\mathbf A_{2h} = \mathbf R \mathbf A_h \mathbf R^T$$ (maybe to within a constant? I forget).